DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
79 
which, by (5.), (15.) and (16.), becomes 
, d( Z) dZ dyi dZ dyi 
D*' dxi dy l dx x 'dy 2 dx 2 '~ "" ’ 
t?(Z) dZ dZ dy x dZ dy 2 
n no . r! nn . * s In ri nn . * rh i n nn . • * * * 
but it is plain that 
(since (Z) would be derived from Z by substituting in the latter the expressions for 
fyi 
dxi 
dxi dxi dy x dxi dy% dx t 
y x ,y . 2 , ... Hyp. III.). And since c ^=hli) See., comparing the two equations last written, 
we obtain 
dZ 
Vi ~ dx- 
(17.) 
The system of 2 ra equations (16.) and (17-) express the result of eliminating the 
2 n constants from the equations (5.) and (11.) and their differential coefficients with 
respect to t. In other words, (16.) and (17-) are a system of 2 n simultaneous differ- 
ential equations of the first order, of which (5.) and (11.), or again, the equations 
supposed in Hyp. I. or II., art. 5, are the 2 n integrals. 
7. There are other remarkable relations between the partial differential coefficients 
of the expressions supposed in Hyp. I. and II., art. 5. For if we differentiate the 
dX 
equation -^-= 6 ; with respect to a i {Hyp. I.), we obtain 
d 2 X . d 2 X dx j d 2 X dx^ 
i — ‘i hi — n 
ddidaj 1 da i dx l dcij ' dd t dx 2 ddj' ' " ’ 
+ 
which gives, putting bj for ^ and y t for 
acij doc i 
dbj dy t dx ^ dy% dx% . ~ yn "-~n 
. ■ nn. nn. • r!n. fin. ' *** ■ fin. fin. 
dy n dx n 
:0 
(a.) 
(b.) 
dat ' ddi daj ' ddi ddj ’ ' * "T" ddi ddj 
^where ^ refers to Hyp. IV., Sec. to Hyp. III., and Sec. to Hyp. 1.^. 
If then this equation be multiplied by ^ {Hyp. II.) and the result summed with 
respect to i, the sum of the first terms is ^ {Hyp. II.), and for the rest, the coeffi- 
cient of ^7 reduces itself to unity, whilst those of the remaining terms vanish (art. 1 , 
equ. ( 1 .), ( 2 .)). Thus we have 
dbj__ 
dyi 
(where the first side refers to Hyp. II., and the second to Hyp. I.). 
Now if we treat the equations 
dY dY 
dx k 
'ddj 
( 18 .) 
